where \(\Gamma ^\rho _{\mu \nu }\)  are the Christoffel symbols for the metric \(g_{\mu \nu }\) . Hereinafter all the quantities associated to the metric \(G_{\mu \nu }\)  are designated by tildas. From (REF ) we see that the connection \(\tilde{\Gamma }^\rho _{\mu \nu }\)  tends to the projected connection (see, e.g., [1]}, [2]}, [3]}, [4]}) when \(\lambda \)  goes to infinity. The curvature tensor for the connection \(\tilde{\Gamma }^\rho _{\mu \nu }\)  is defined in the standard way
\(\tilde{R}^\rho _{\ \nu \sigma \mu }=R^\rho _{\ \nu \sigma \mu }+\nabla _{[\sigma }\gamma _{\mu ]\nu }^\rho +\gamma _{[\sigma \lambda }^\rho \gamma _{\mu ]\nu }^\lambda .\) 
